[Une borne de type moyenne harmonique pour le trou spectral du laplacien sur les variétés riemanniennes]
La plupart des minorants connus pour le trou spectral du laplacien faisant intervenir la courbure de Ricci sont basés sur l'infimum de cette courbure, et peuvent être de piètre qualité si la courbure de Ricci est élevée partout sauf sur un petit sous-ensemble sur lequel elle est faible. On montre ici que la moyenne harmonique de la courbure de Ricci est un minorant du trou spectral du laplacien, ce qui résout partiellement le problème (malheureusement, il faut supposer que la courbure de Ricci est partout positive ou nulle).
Most known lower bounds on the spectral gap of the Laplacian using Ricci curvature are based on the infimum of the Ricci curvature, and can be really poor when the Ricci curvature is large everywhere but on a small subset on which it is small. Here we show that the harmonic mean of the Ricci curvature is a lower bound on the spectral gap of the Laplacian, which partially solves the problem (unfortunately, we have to assume that the Ricci curvature is everywhere nonnegative).
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@article{CRMATH_2010__348_23-24_1319_0,
author = {Veysseire, Laurent},
title = {A harmonic mean bound for the spectral gap of the {Laplacian} on {Riemannian} manifolds},
journal = {Comptes Rendus. Math\'ematique},
pages = {1319--1322},
publisher = {Elsevier},
volume = {348},
number = {23-24},
year = {2010},
doi = {10.1016/j.crma.2010.10.015},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2010.10.015/}
}
TY - JOUR AU - Veysseire, Laurent TI - A harmonic mean bound for the spectral gap of the Laplacian on Riemannian manifolds JO - Comptes Rendus. Mathématique PY - 2010 SP - 1319 EP - 1322 VL - 348 IS - 23-24 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2010.10.015/ DO - 10.1016/j.crma.2010.10.015 LA - en ID - CRMATH_2010__348_23-24_1319_0 ER -
%0 Journal Article %A Veysseire, Laurent %T A harmonic mean bound for the spectral gap of the Laplacian on Riemannian manifolds %J Comptes Rendus. Mathématique %D 2010 %P 1319-1322 %V 348 %N 23-24 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2010.10.015/ %R 10.1016/j.crma.2010.10.015 %G en %F CRMATH_2010__348_23-24_1319_0
Veysseire, Laurent. A harmonic mean bound for the spectral gap of the Laplacian on Riemannian manifolds. Comptes Rendus. Mathématique, Tome 348 (2010) no. 23-24, pp. 1319-1322. doi : 10.1016/j.crma.2010.10.015. http://www.numdam.org/articles/10.1016/j.crma.2010.10.015/
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